Yes, this is possible. For instance, consider the opposite category of nonempty sets. A singleton set $i$ is initial in this category (since it is terminal in the usual category of sets), but there is a map from every object to $i$ (since there is a function from $i$ to any nonempty set). And $i$ is not terminal, since if $a$ is any set with more than one element then there is more than one map $a\to i$.
However, if you assume that your category has a terminal object $t$ and an initial object $i$ with a map from every object, then $i$ must be terminal. For if $f:t\to i$ is any map, let $g:i\to t$ be the unique map (using the universal property of either $i$ or $t$). Then $fg$ and $gf$ must both be the identity, because there is only one map $i\to i$ and only one map $t\to t$. So $f$ and $g$ are inverse isomorphisms, so $i$ is terminal since $t$ is.